The advantage of truncated permutations

Constructing a Pseudo Random Function (PRF) is a fundamental problem in cryptology. Such a construction, implemented by truncating the last m bits of permutations of {0,1}ⁿ was suggested by Hall et al. (1998). They conjectured that the distinguishing advantage of an adversary with q queries, Adv_n,m(q), is small if q=o(2^(n+m)∕2), established an upper bound on Adv_n,m(q) that confirms the conjecture for m<n∕7, and also declared a general lower bound Adv_n,m(q)=Ω(q²∕2^n+m). The conjecture was essentially confirmed by Bellare and Impagliazzo (1999). Nevertheless, the problem of estimating Adv_n,m(q) remained open. Combining the trivial bound 1, the birthday bound, and a result of Stam (1978) leads to the upper bound [Formula presented] In this paper we show that this upper bound is tight for every 0≤m<n and any q. This, in turn, verifies that the converse to the conjecture of Hall et al. is also correct, i.e., that Adv_n,m(q) is negligible only for q=o(2^(n+m)∕2).